Figure 5

Effect of Gaussian coherent noise on the output of the parametrized quantum circuit shown in Fig. 2(b). The plot is obtained by first choosing a parameter vector \(\boldsymbol{\theta}_{0} \in \mathbb{R}^{92}\) corresponding to a the ideal noise-free expectation value \(f(\boldsymbol{\theta}_{0})=\langle O\rangle\) with \(O = Z^{\otimes 4}\). With this baseline fixed, random Gaussian perturbations are added to the angles \(\boldsymbol{\theta}_{noisy} = \boldsymbol{\theta}_{0} + \delta \boldsymbol{\theta}\), and the resulting noisy expectation vales \(\langle O\rangle_{noisy}\) are computed. Each point in the plot is the average over \(N=10^{5}\) different perturbation vectors sampled from a multivariate Gaussian distribution of a given σ. The experiments are then repeated for increasing values of the noise strength σ. The error bars show the statistical error of the mean. For small noise levels, the output of the quantum circuit closely follows the behaviour predicted by Equation (20), where the Hessian is evaluated at the unperturbed value \(H=H(\boldsymbol{\theta}_{0})\). When the error is too large the circuit behaves as a random circuit whose output is on average zero, hence the error plateaus to the unperturbed expectation value \(\varepsilon = |\langle O\rangle| = |f(\boldsymbol{\theta}_{0})| \)). The upper bound predicted by Equation (25) is very loose in general, and holds tightly only for very small values of \(\sigma \lessapprox 0.01\)